A Study of the Skein Module of the Figure-Eight Knot Complement
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This dissertation studies the action of the Reshetikhin–Turaev skein alge- bra of the torus on the Reshetikhin–Turaev skein module of the figure-eight knot complement. The first chapter contains the historical background on the Reshetikhin– Turaev skein module, the Jones polynomial, and the colored Jones polynomials. The second chapter exhibits the multiplication rule for the Rashetikhin– Turaev skein algebra of the cylinder over the torus, which has appeared in the work of R. Gelca and A. Uribe, and is an analogue of the formula for the Kauffman bracket skein modules, which was found by Ch. Frohman and R. Gelca. In the third and fourth chapters we perform several computations with skeins in the figure-eight knot complement in order to obtain several formulas rel- evant in the subsequent chapters. We compare these formulas with those obtained by R. Gelca and J. Sain using the Kauffman bracket. The fifth chapter consists of explicit formulas for the action of (1, 0) T on the basis elements of the Reshetikhin–Turaev skein module of the knot complement. In the sixth chapter we describe the action of the Reshetikhin–Turaev skein algebra of the torus on the Reshetikhin–Turaev skein module of the figure-eight knot complement using the results obtained in the fifth chapter and describe how we arrive at the colored Jones polynomials. The fifth and the sixth chapters contain the main results of this dissertation.